Singular value decomposition of the third multivariate moment

The third moment of a random vector is a matrix which conveniently arranges all moments of order three which can be obtained from the random vector itself. We investigate some properties of its singular value decomposition. In particular, we show that left eigenvectors corresponding to positive singular values of the third moment are vectorized, symmetric matrices. We derive further properties under the additional assumptions of exchangeability, reversibility and independence. Statistical applications deal with measures of multivariate skewness.